Upper Tail Large Deviations for Arithmetic Progressions in a Random Set

Bhattacharya, Bhaswar B.; Ganguly, Shirshendu; Shao, Xuancheng; Zhao, Yufei

doi:10.1093/imrn/rny022

Cited by 21 publications

(33 citation statements)

References 34 publications

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“…The rate at which p is allowed to decay for (1) to hold turns out to depend on Gaussian widths of the form featuring in Theorem 1.1. The bounds proved in [BGSZ18] imply that (1)…”

Section: It Was Proved Independently By Frantzikinakis Et Al [Flw12]mentioning

confidence: 99%

“…, x n ] is the mapping ∇p : R n → R n whose ith coordinate is given by (∇p) i = (∂p/∂x i )(x). The proof of Theorem 1.4 follows from a simple corollary of Theorem 1.1 and one of the main results of [BGSZ18]. For the corollary, we consider polynomial mappings given by gradients of polynomials of the form (3).…”

Section: Upper Tails For Arithmetic Progressions In Random Setsmentioning

confidence: 99%

“…The main motivation for finding such precise estimates of the upper tail probability is not so much the problem itself as it is to understand structure of the set [Z/N Z] p conditioned on X k being much larger than its expectation (see[BGSZ18]). …”

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confidence: 99%

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Gaussian Width Bounds with Applications to Arithmetic Progressions in Random Settings

Briët

Gopi

2018

International Mathematics Research Notices

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Motivated by a problem on random differences in Szemerédi's theorem and another problem on large deviations for arithmetic progressions in random sets, we prove upper bounds on the Gaussian width of special point sets in R k . The point sets are formed by the image of the n-dimensional Boolean hypercube under a mapping ψ : R n → R k , where each coordinate is a constant-degree multilinear polynomial with 0-1 coefficients. We show the following applications of our bounds. Let [Z/N Z] p be the random subset of Z/N Z containing each element independently with probability p.This gives a polynomial improvement for all ℓ ≥ 3 of a previous bound due to Frantzikinakis, Lesigne and Wierdl, and reproves more directly the same improvement shown recently by the authors and Dvir (here we avoid the theories of locally decodable codes and quantum information).• Let X k be the number of k-term arithmetic progressions in [Z/N Z] p and consider the large deviation rate ρ k (δ) = log Pr[X k ≥ (1 + δ)EX k ]. We give quadratic improvements of the best-known range of p for which a highly precise estimate of ρ k (δ) due to Bhattacharya, Ganguly, Shao and Zhao is valid for all odd k ≥ 5. In particular, the estimate holds if p ≥ ω(N −c k log N ) for c k = (6k⌈(k − 1)/2⌉) −1 .We also discuss connections with locally decodable codes and the Banach-space notion of type for injective tensor products of ℓ p -spaces.

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“…The rate at which p is allowed to decay for (1) to hold turns out to depend on Gaussian widths of the form featuring in Theorem 1.1. The bounds proved in [BGSZ18] imply that (1)…”

Section: It Was Proved Independently By Frantzikinakis Et Al [Flw12]mentioning

confidence: 99%

Section: Upper Tails For Arithmetic Progressions In Random Setsmentioning

confidence: 99%

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Gaussian Width Bounds with Applications to Arithmetic Progressions in Random Settings

Briët

Gopi

2018

International Mathematics Research Notices

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“…The corresponding problem for the lower tail was studied in [31], where several questions remain open. The analogous problem for the number the arithmetic progressions in a random set was recently studied in [8].…”

Section: Introductionmentioning

confidence: 99%

Upper Tails for Edge Eigenvalues of Random Graphs

Bhattacharya¹,

Ganguly²

2020

SIAM J. Discrete Math.

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In this note we prove a precise large deviation principle for the largest and second largest eigenvalues of a sparse Erdős-Rényi graph. Our arguments rely on various recent breakthroughs in the study of mean field approximations for large deviations of low complexity non-linear functions of independent Bernoulli variables and solutions of the associated entropic variational problems.1 1 Note that A(Gn,p) − p11 ′ is the centered adjacency matrix up to a translation by pI, where I is the identity matrix. Since this deterministically shifts every eigenvalue by p which is asymptotically negligible, we will continue this abuse of terminology.2 We write f g to denote f = O(g), f ∼ g means f = (1 + o(1))g, and f ≪ g means f = o(g).

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“…Complementing these results, Bhattacharya, Ganguly, Shao, and Zhao [2] pinned down the precise large deviation rate function for "sufficiently large" p. By contrast to the approach in [27], the proof in [2] builds on the non-linear large deviation principle by Chatterjee and Dembo [8] and its refinement due to Eldan [11] in terms of the concept of Gaussian width, a particular notion of complexity. Recently, Briët and Gopi [6] derived an upper bound on the Gaussian width leading to an improvement of the lower bound on p given in [2]. The special case ℓ = 3 was already included in [8].…”

Section: Related Workmentioning

confidence: 98%

Bivariate fluctuations for the number of arithmetic progressions in random sets

Barhoumi-Andréani¹,

Koch²,

Liu³

2019

Electron. J. Probab.

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We study arithmetic progressions {a, a+b, a+2b, . . . , a+(ℓ−1)b}, with ℓ ≥ 3, in random subsets of the initial segment of natural numbers [n] := {1, 2, . . . , n}. Given p ∈ [0, 1] we denote by [n]p the random subset of [n] which includes every number with probability p, independently of one another. The focus lies on sparse random subsets, i.e. when p = p(n) = o(1) as n → +∞.Let X ℓ denote the number of distinct arithmetic progressions of length ℓ which are contained in [n]p. We determine the limiting distribution for X ℓ not only for fixed ℓ ≥ 3 but also when ℓ = ℓ(n) → +∞. The main result concerns the joint distribution of the pair (X ℓ , X ℓ ′ ), ℓ > ℓ ′ , for which we prove a bivariate central limit theorem for a wide range of p. Interestingly, the question of whether the limiting distribution is trivial, degenerate, or nontrivial is characterised by the asymptotic behaviour (as n → +∞) of the threshold function ψ ℓ = ψ ℓ (n) := np ℓ−1 ℓ. The proofs are based on the method of moments and combinatorial arguments, such as an algorithmic enumeration of collections of arithmetic progressions.

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Upper Tail Large Deviations for Arithmetic Progressions in a Random Set

Cited by 21 publications

References 34 publications

Gaussian Width Bounds with Applications to Arithmetic Progressions in Random Settings

Gaussian Width Bounds with Applications to Arithmetic Progressions in Random Settings

Upper Tails for Edge Eigenvalues of Random Graphs

Bivariate fluctuations for the number of arithmetic progressions in random sets

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